Advertisements
Advertisements
Question
Discuss the continuity of the f(x) at the indicated points:
(i) f(x) = | x | + | x − 1 | at x = 0, 1.
Solution
Given:
We have
(LHL at x = 0) =
(RHL at x = 0) =
Also
Now,
(LHL at x = 1) =
(RHL at x =1) =
Also,
Hence,
APPEARS IN
RELATED QUESTIONS
Determine the value of 'k' for which the following function is continuous at x = 3
`f(x) = {(((x + 3)^2 - 36)/(x - 3), x != 3), (k, x = 3):}`
Examine the following function for continuity:
f(x) = | x – 5|
Discuss the continuity of the function f, where f is defined by `f(x) = {(3, ","if 0 <= x <= 1),(4, ","if 1 < x < 3),(5, ","if 3 <= x <= 10):}`
Discuss the continuity of the function f, where f is defined by `f(x) = {(-2,"," if x <= -1),(2x, "," if -1 < x <= 1),(2, "," if x > 1):}`
If \[f\left( x \right) = \begin{cases}e^{1/x} , if & x \neq 0 \\ 1 , if & x = 0\end{cases}\] find whether f is continuous at x = 0.
Discuss the continuity of the following functions at the indicated point(s):
(ii) \[f\left( x \right) = \left\{ \begin{array}{l}x^2 \sin\left( \frac{1}{x} \right), & x \neq 0 \\ 0 , & x = 0\end{array}at x = 0 \right.\]
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \binom{\frac{x^3 + x^2 - 16x + 20}{\left( x - 2 \right)^2}, x \neq 2}{k, x = 2}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\left| x - 3 \right|, & \text{ if } x \geq 1 \\ \frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}, & \text{ if } x < 1\end{cases}\]
Discuss the continuity of the function \[f\left( x \right) = \begin{cases}2x - 1 , & \text { if } x < 2 \\ \frac{3x}{2} , & \text{ if } x \geq 2\end{cases}\]
If \[f\left( x \right) = \begin{cases}\frac{x^2 - 16}{x - 4}, & \text{ if } x \neq 4 \\ k , & \text{ if } x = 4\end{cases}\] is continuous at x = 4, find k.
Let \[f\left( x \right) = \begin{cases}\frac{x^4 - 5 x^2 + 4}{\left| \left( x - 1 \right) \left( x - 2 \right) \right|}, & x \neq 1, 2 \\ 6 , & x = 1 \\ 12 , & x = 2\end{cases}\]. Then, f (x) is continuous on the set
If \[f\left( x \right) = \begin{cases}\frac{\sin (a + 1) x + \sin x}{x} , & x < 0 \\ c , & x = 0 \\ \frac{\sqrt{x + b x^2} - \sqrt{x}}{bx\sqrt{x}} , & x > 0\end{cases}\]is continuous at x = 0, then
The value of f (0) so that the function
If \[f\left( x \right) = \frac{1}{1 - x}\] , then the set of points discontinuity of the function f (f(f(x))) is
The values of the constants a, b and c for which the function \[f\left( x \right) = \begin{cases}\left( 1 + ax \right)^{1/x} , & x < 0 \\ b , & x = 0 \\ \frac{\left( x + c \right)^{1/3} - 1}{\left( x + 1 \right)^{1/2} - 1}, & x > 0\end{cases}\] may be continuous at x = 0, are
The points of discontinuity of the function
\[f\left( x \right) = \begin{cases}2\sqrt{x} , & 0 \leq x \leq 1 \\ 4 - 2x , & 1 < x < \frac{5}{2} \\ 2x - 7 , & \frac{5}{2} \leq x \leq 4\end{cases}\text{ is } \left( \text{ are }\right)\]
If \[f\left( x \right) = \begin{cases}\frac{1 - \sin^2 x}{3 \cos^2 x} , & x < \frac{\pi}{2} \\ a , & x = \frac{\pi}{2} \\ \frac{b\left( 1 - \sin x \right)}{\left( \pi - 2x \right)^2}, & x > \frac{\pi}{2}\end{cases}\]. Then, f (x) is continuous at \[x = \frac{\pi}{2}\], if
Show that f(x) = |x − 2| is continuous but not differentiable at x = 2.
Show that \[f\left( x \right) =\]`{(12x, -,13, if , x≤3),(2x^2, +,5, if x,>3):}` is differentiable at x = 3. Also, find f'(3).
Write the points of non-differentiability of
The set of points where the function f (x) = x |x| is differentiable is
Let f (x) = |sin x|. Then,
Find k, if the function f is continuous at x = 0, where
`f(x)=[(e^x - 1)(sinx)]/x^2`, for x ≠ 0
= k , for x = 0
If the function f is continuous at = 2, then find f(2) where f(x) = `(x^5 - 32)/(x - 2)`, for ≠ 2.
If the function f is continuous at x = 0
Where f(x) = 2`sqrt(x^3 + 1)` + a, for x < 0,
= `x^3 + a + b, for x > 0
and f (1) = 2, then find a and b.
The total cost C for producing x units is Rs (x2 + 60x + 50) and the price is Rs (180 - x) per unit. For how many units the profit is maximum.
If f(x) = `(e^(2x) - 1)/(ax)` . for x < 0 , a ≠ 0
= 1. for x = 0
= `(log(1 + 7x))/(bx)`. for x > 0 , b ≠ 0
is continuous at x = 0 . then find a and b
If f (x) = `(1 - "sin x")/(pi - "2x")^2` , for x ≠ `pi/2` is continuous at x = `pi/4` , then find `"f"(pi/2) .`
Discuss the continuity of the function `f(x) = (3 - sqrt(2x + 7))/(x - 1)` for x ≠ 1
= `-1/3` for x = 1, at x = 1
Discuss the continuity of the function f at x = 0, where
f(x) = `(5^x + 5^-x - 2)/(cos2x - cos6x),` for x ≠ 0
= `1/8(log 5)^2,` for x = 0
Discuss the continuity of the function f(x) = sin x . cos x.
The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at ______.
The number of points at which the function f(x) = `1/(x - [x])` is not continuous is ______.
y = |x – 1| is a continuous function.
f(x) = `{{:(("e"^(1/x))/(1 + "e"^(1/x))",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0
The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is ______.
An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is ______.
The composition of two continuous function is a continuous function.
If the following function is continuous at x = 2 then the value of k will be ______.
f(x) = `{{:(2x + 1",", if x < 2),( k",", if x = 2),(3x - 1",", if x > 2):}`
